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# ℝ^{3}

*Journal of Inequalities and Applications*
**volume 2012**, Article number: 3 (2012)

## Abstract

A special case of Mahler volume for the class of symmetric convex bodies in ℝ^{3} is treated here. It is shown that a cube has the minimal Mahler volume and a cylinder has the maximal Mahler volume for all generalized cylinders. Further, the Mahler volume of bodies of revolution obtained by rotating the unit disk in ℝ^{2} is presented.

**2000 Mathematics Subject Classification:** 52A20; 52A40.

## 1 Introduction

Throughout this article a convex body *K* in Euclidean *n*-space ℝ^{n} is a compact convex set that contains the origin in its interior. Its polar body *K** is defined by

where *x*·*y* denotes the standard inner product of *x* and *y* in ℝ^{n}.

If *K* is an origin symmetric convex body, then the product

is called the volume product of *K*, where *V* (*K*) denotes *n*-dimensional volume of *K*, which is known as the *Mahler volume* of *K*, and it is invariant under linear transformation.

One of the main questions still open in convex geometric analysis is the problem of finding a sharp lower estimate for the Mahler volume of a convex body *K* (see the survey article [1]).

A sharp upper estimate of the volume product is provided by the Blaschke-Santaló inequality: For every centered convex body *K* in ℝ^{n}

with equality if and only if *K* is an ellipsoid centered at the origin, where *ω*_{
n
} is the volume of the unit ball in ℝ^{n} (see, e.g., [2–5]).

The Mahler conjecture for the class of origin-symmetric bodies is that:

with equality holding for parallelepipeds and their polars. For *n* = 2, the inequality is proved by Mahler himself [6], and in 1986, Reisner [7] showed that parallelograms are the only minimizers. Reisner [8] established inequality (1.1) for a class of bodies that have a high degree of symmetry, known as zonoids, which are limits of finite Minkowski sums of line segments. Lopez and Reisner [9] proved the inequality (1.1) for *n* ≤ 8 and the minimal bodies are characterized. Recently, Nazaeov et al. [10] proved that the cube is a strict local minimizer for the Mahler volume in the class of origin-symmetric convex bodies endowed with the Banach-Mazur distance.

Bourgain and Milman [11] have proved that there exists a constant *c >* 0 independent of the dimension *n*, such that for all origin-symmetric bodies *K*,

which is now known as the reverse Santaló inequality. Recently, Kuperberg [12] found a beautiful new approach to the reverse Santaló inequality. What's especially remarkable about Kuperberg's inequality is that it provides an explicit value for *c*. However, the Mahler conjecture is still open even in the three-dimensional case, Tao [13] made an excellent remark about the open question.

In the present article, we treat a special case of Mahler volume in ℝ^{3}. We now introduce some notations: A real-valued function *f*(*x*) is called *concave*, if for any *x*, *y* ∈ [*a*, *b*] and any *λ* ∈ [0, 1], they have

**Definition 1** *In three-dimensional Cartesian coordinate system OXYZ, if C*′ *is an origin-symmetric convex body in coordinate plane YOZ, then the set:*

*is defined as a generalized cylinder in* ℝ^{3}.

**Definition 2**
*In the coordinate plane XOY, let*

*where f*(*x*) *(*[*-a*, *a*]*, a >* 0*), is a nonnegative concave and even function. Rotating D about the X-axis in* ℝ^{3}*, we can get a geometric object*

*We define the geometric object R as a body of revolution generated by the function f*(*x*) *(or by the domain D), and call the function f*(*x*) *as the generated function of R and D as the generated domain of R*.

If the generated domain of *R* is a rectangle and a diamond, *R* is called a *cylinder* and a *bicone*, respectively.

Let *C* denotes the set of all generalized cylinders. In this article, we proved that among the generalized cylinders, a cube has the minimal Mahler volume and a cylinder has the maximal Mahler volume, theorem as following:

**Theorem 1** *For* C\in \mathcal{C}, *we have*

*where C*_{0} = [-1, 1] × [-1, 1] × [-1, 1] *is a cube and C*_{1} = [-1, 1] × *B*^{2} *is cylinder*.

Further, we get the following theorem:

**Theorem 2**
*For a class of bodies of revolution obtained by rotating the "unit disk" in planar XOY, where the "unit disk" is the following set:*

*the Mahler volume is increasing for* 1 ≤ *p* ≤ 2 *and decreasing for* 2 ≤ *p* ≤ +∞.

More interrelated notations, definitions, and their background materials are exhibited in the following section.

## 2 Definition and notation

The setting for this article is *n*-dimensional Euclidean space ℝ^{n}. Let {\mathcal{K}}^{n} denotes the set of convex bodies (compact, convex subsets with non-empty interiors), {\mathcal{K}}_{o}^{n} denotes the subset of {\mathcal{K}}^{n} that contains the origin in their interiors. As usual, *B*^{n} denotes the unit ball centered at the origin, *S*^{n-1}the unit sphere, *o* the origin, and ||·|| the norm in ℝ^{n}.

If *u* ∈ *S*^{n-1}is a direction, *u*^{⊥} is the (*n -* 1)-dimensional subspace orthogonal to *u*. For *x*, *y* ∈ ℝ^{n}, *x*·*y* is the inner product of *x* and *y*, and [*x*, *y*] denotes the line segment with endpoints *x* and *y*.

If *K* is a set, ∂*K* is its boundary, *int K* is its interior, and *conv K* denotes its convex hull. *V* (*K*) denotes *n*-dimensional volume of *K*. Let *K*|*S* be the orthogonal projection of *K* into a subspace *S*.

Let K\in {\mathcal{K}}^{n} and *H* = {*x* ∈ ℝ^{n}|*x·v* = *d*} denotes a hyperplane, *H*^{+} and *H*^{-} denote the two closed halfspaces bounded by *H*.

Associated with each convex body *K* in ℝ^{n}, its *support function h*_{
K
} : ℝ^{n} *-* [0, ∞), is defined for *x* ∈ ℝ^{n}, by

and its *radial function ρ*_{
K
} : ℝ^{n}*\*{0} → (0, ∞), is defined for *x* ≠ 0, by

From the definitions of the support and radial functions and the definition of the polar bodies, it follows that (see [4])

If *P* is a polytope, i.e., *P* = conv{*p*_{1}, ..., *p*_{
m
}}, where *p*_{
i
} (*i* = 1, ..., *m*) are vertices of polytope *P*. By the definition of polar body, we have

which implies that *P** is the intersection of *m* closed halfspace with exterior normal vector *p*_{
i
} and the distance of hyperplane {*x* ∈ ℝ^{n} : *x*·*p*_{
i
} = 1} from the origin is 1*/*||*p*_{
i
}||.

For K\in {\mathcal{K}}_{o}^{n}, if *x* = (*x*_{1}, *x*_{2}, ..., *x*_{
n
}) ∈ *K*, *x*′ = (*ε*_{1}*x*_{1}, ..., *ε*_{
n
}*x*_{
n
}) ∈ *K* for any signs *ε*_{
i
} = ± 1 (*i* = 1, ..., *n*), then *K* is a *1-unconditional convex body*. In fact, *K* is symmetric around all coordinate hyperplanes.

To proof the inequality, we give the following definitions.

**Definition 3**
*In Definition 2, if the function*

*where k and b are real constants, and f*(*-a*) = 0*, then the body of revolution is defined as a bicone. In three-dimensional Cartesian coordinate system OXYZ, if C*′ *is an origin-symmetric convex body in coordinate plane YOZ and points A* = (*-a*, 0, 0) *and A*′ = (*a*, 0, 0)*, then the set*.

*is defined as a generalized bicone in* ℝ^{3}.

## 3 Proof of the main results

In this section, we only consider convex bodies in three-dimensional Cartesian coordinate system with origin *O*, and its three coordinate axes are denoted by *X*-, *Y* -, and *Z*-axis.

Let *C* be a generalized cylinder as following:

where *C*′ is an origin-symmetric convex body in coordinate plane *Y OZ*.

We require the following lemmas to prove our main result.

**Lemma 1** *If* K\in {\mathcal{K}}_{o}^{3}*, for any u*∈ *S*^{2}*, then*

*On the other hand, if*
{K}^{\prime}\in {\mathcal{K}}_{o}^{3}
*satisfies*

*for any* u\in {S}^{2}\cap {v}_{0}^{\perp} (*v*_{0} *is a fixed vector*)*, then*

**Proof** First, we prove (3.1).

Let *x* ∈ *u*^{⊥}, *y* ∈ *K* and *y*′ = *y*|*u*^{⊥}, since the hyperplane *u*^{⊥} is orthogonal to the vector *y* - *y*′, then

If *x* ∈ *K** | *u*^{⊥}, for any *y*′ ∈ *K* | *u*^{⊥}, there exists *y* ∈ *K* such that *y*′ = *y*|*u*^{⊥}, then *x·y*′ = *x*·*y* ≤ 1, and *x* ∈ (*K*|*u*^{⊥})*. Hence,

If *x* ∈ (*K*|*u*^{⊥})*, then for any *y* ∈ *K* and *y*′ = *y*|*u*^{⊥}, *x*·*y* = *x*·*y*′ *≤* 1, thus *x* ∈ *K**, and since *x* ∈ *u*^{⊥}, thus *x* ∈ *K** | *u*^{⊥}. Then,

Next, we prove (3.2).

Let {S}^{1}={S}^{2}\cap {v}_{0}^{\perp}. For any direction vector *v* ∈ *S*^{2}, there always exists a *u* ∈ *S*^{1} satisfying *v* ∈ *u*^{⊥}. Since

and by (3.1)

thus

Then, we get

By the arbitrary of direction *v*, we obtain the desired result. ■

For any C\in \mathcal{C} and any *u* ∈ *B*^{2} | *v*^{⊥} (*v* = (1, 0, 0)), *C*|*u*^{⊥} is a rectangle by the above definition. We study the polar body of a rectangle in the planar. From Figure 1, if *C*|*u*^{⊥} = [-1, 1]*×*[*-a*, *a*], its polar body in planar *XOY* is a diamond (vertices are (-1, 0), (1, 0), (0, -1*/a*), (0, 1*/a*)), thus we can get the following Lemma 2.

**Lemma 2**
*For any*
C\in \mathcal{C}
*, if*

*where C*′ *is an origin-symmetric convex body in coordinate plane YOZ, then C** *is a generalized bicone with vertices* (-1, 0, 0) *and* (1, 0, 0) *and the base* (*C*′)*.

**Proof** Let *v*_{0} = (1, 0, 0), {S}^{1}={S}^{2}\cap {v}_{0}^{\perp}. By Lemma 1, we have

for any *u* ∈ *S*^{1}. Because that (*C*|*u*^{⊥})* is a diamond with vertices (-1, 0, 0) and (1, 0, 0), *C** | *u*^{⊥} is a diamond with vertices (-1, 0, 0) and (1, 0, 0) for any *u* ∈ *S*^{1}, which implies that *C** is a bicone with vertices (-1, 0, 0) and (1, 0, 0).

In view of

and C|{v}_{0}^{\perp}={C}^{\prime}, then, the base of *C** is (*C*′)*. ■

In the following, we will restate and prove Theorem 1.

**Theorem 1**
*For*
C\in \mathcal{C}
*, we have*

*where C*_{0} = [-1, 1] *×* [-1, 1] *×* [-1, 1] *is a cube and C*_{1} = [-1, 1] *× B*^{2} *is cylinder*.

**Proof** Let *v* = (1, 0, 0), and *V* (*C*) = *V* (*C*_{0}) by linear transformation, thus *V* (*C* ∩ *v*⊥) = *V* (*C*_{0} ∩ *v*^{⊥}).

In planar *v*^{⊥}, since the square has the minimal Mahler volume in ℝ^{2}, thus

we get

then

where the equality holds if and only if *C* ∩ *v*^{⊥} is a square. Hence,

Similarly, let *V* (*C*) = *V* (*C*_{1}) for any C\in \mathcal{C} by linear transformation, then *V* (*C* ∩ *v*^{⊥}) = *V* (*C*_{1} ∩ *v*^{⊥}).

Since *C*_{1} ∩ *v*^{⊥} is a disk, which has the maximal Mahler volume in ℝ^{2}, thus

we get

Hence, V\left({C}_{1}^{*}\right)\ge V\left({C}^{*}\right), which implies

■

Theorem 1 implies that among the generalized cylinders, a cube has the minimal Mahler volume and a cylinder has the maximal Mahler volume.

## 4 Mahler volume of a special class of bodies of revolution

In this section, we study a special case in the coordinate plane *XOY*, and define the "unit disk" in planar *XOY* as following set:

We need the following lemmas to prove our result.

**Lemma 3** *Let P is a 1-unconditional convex body and P** *is its polar body in the coordinate plane XOY. Let R and R*′ *are two bodies of revolution obtained by rotating P and P***, respectively. Then R*′ = *R**.

**Proof** Let *v*_{0} = {1, 0, 0} and {S}^{1}={S}^{2}\cap {v}_{0}^{\perp}, for any *u* ∈ *S*^{1}, we have

Since *R*′ ∩ *u*⊥ = (*R* ∩ *u*^{⊥})* for any *u* ∈ *S*^{1}, we get

for any *u* ∈ *S*^{1}. By Lemma 1, we have *R*′ = *R**. ■

**Lemma 4**
*If*

*then the polar body of*

*is the following set:*

**Proof** For any (*x*, *y*) ∈ *U* and (*x*′, *y*′) ∈ *U*′, we have

which implies *U*′ ⊂ *U**.

If a point *A*′ = (*x*′, *y*′) ∉ *U*′, then

Let {A}_{0}^{\prime}=\left(\left|{x}^{\prime}\right|,\left|{y}^{\prime}\right|\right), then {A}_{0}^{\prime}\notin {U}^{\prime}. There exists a real *r >* 1 and a point *A*^{0} ∈ ∂*U*′ satisfying {A}_{0}^{\prime}=r{A}^{0}. If *A*^{0} = (*x*_{0}, *y*_{0}), then *x*_{0} *>* 0 and *y*_{0} *>* 0. Let x={x}_{0}^{\frac{q}{p}} and y={y}_{0}^{\frac{q}{p}}, then

and

which implies (*x*, *y*) ∈ *U* and 〈(*x*, *y*), (|*x*′|, |*y*′|)〉 = *r >* 1, thus {A}_{0}^{\prime}\notin {U}^{*}. Because that *U** is a 1-unconditional convex body, we have *A*′ ∉ *U**. Then, *U** ⊂ *U*′. ■

Rotating *U* and *U*′, we can get two bodies of revolution *R* and *R*′. By Lemma 3, we have *R*′ = *R**. Let *F*(*p*) = *V* (*R*)*V* (*R**).

In the following, we restate and prove Theorem 2.

**Theorem 2**
*For a class of bodies of revolution obtained by rotating the "unit disk" in planar XOY, where the "unit disk" is the following set:*

*the Mahler volume is increasing for* 1 ≤ *p* ≤ 2 *and decreasing for* 2 ≤ *p* ≤ +∞.

**Proof** By integration, we get *V*_{
R
}(*p*) and *V*_{R*}(*q*), which are volume functions of *R* and *R** about *p* and *q* as following:

and

Thus, we have the Mahler volume *V* (*R*)*V* (*R**), which is a function about *p* as following:

where *p* ≥ 1, and \frac{1}{p}+\frac{1}{q}=1.

Let 1 - *x*^{p} = *y*, we have

where *B*(·, ·) is Beta function. Thus we have

where *p* ≥ 1, and \frac{1}{p}+\frac{1}{q}=1.

By the relationship between Gamma function and Beta function:

we have

And by the following properties of Gamma function:

we have

We can easily prove

then *R* and *R** are bicone and cylinder, or cylinder and bicone, and

then *R* and *R** are the same unit ball, which have the maximal Mahler volume.

In fact, *F*(*p*) = *F*(*q*) holds when \frac{1}{p}+\frac{1}{q}=1, so we just need to prove *F* (*p*) is increasing when 1 ≤ *p* ≤ 2, which can be easily proved by *F*′(*p*) ≥ 0 when 1 ≤ *p* ≤ 2. Based on the above conclusions, we have that a cylinder has the minimal Mahler volume and a ball has the maximal Mahler volume in this special class of bodies of revolution. ■

We can draw the figure of the function *F*(*p*) by using MATLAB (see Figure 2). From the figure, we see that function *F*(*p*) is increasing when 1 ≤ *p* ≤ 2 and decreasing when 2 ≤ *p* ≤ +∞, so *F*(2) is a maximum and *F*(1) = *F*(+∞) is a minimum.

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## Acknowledgements

The author express her deep gratitude to the referees for their many very valuable suggestions and comments. The research of Hu-Yan was supported by National Natural Science Foundation of China (10971128), Shanghai Leading Academic Discipline Project (S30104), and the research of Hu-Yan was partially supported by Innovation Program of Shanghai Municipal Education Commission (10yz160).

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Yan, H. ℝ^{3}.
*J Inequal Appl* **2012, **3 (2012). https://doi.org/10.1186/1029-242X-2012-3

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DOI: https://doi.org/10.1186/1029-242X-2012-3

### Keywords

- Mahler volume
- convex body
- polar body
- body of revolution